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Velocity and acceleration algorithm

  1. Apr 29, 2013 #1
    1. The problem statement, all variables and given/known data

    Why are (2) and (4) equations more accurately than (1) and (3) ? Why is 7*dt in (4) equation ? What kind of equations are (2) and (4) ? What method they used to write (3) and (4) equations?

    2. Relevant equations

    Velocity:
    (1) v = (x - x[i-1]) / (t - t[i-1])
    (2) v = (x[i+1] - x[i-1]) / (2*dt)

    Acceleration:
    (3) a = (v - v[i-1]) / (t - t[i-1])
    (4) a = (2*x[i+2] - x[i+1] - 2*x - x[i-1] + 2*x[i-2]) / (7*dt)

    3. The attempt at a solution
    Verlet algorithm
    Finite difference
     
  2. jcsd
  3. Apr 30, 2013 #2

    haruspex

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    The problem with (1) and (3) is that there's a time shift between the input positions and the computed velocities and accelerations. E.g (1) is really estimating the velocity at step i-0.5. (2) corrects that.
    (4) achieves the same result, but I'm not sure why it's quite as it is. (Shouldn't it have dt2 at the end?) If you start with the 'smoothed' acceleration expression (2*ai+1+3*ai+2*ai-1)/7 and then substitute for those ai using the forms ai.Δt = vi+.5-vi-.5 and vi.Δt = xi+.5-xi-.5 you arrive at (4) (with dt2). But why start with (2*ai+1+3*ai+2*ai-1)/7 rather than e.g. (ai+1+2*ai+ai-1)/4 I don't know.
     
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